Properties

Label 2-6026-1.1-c1-0-55
Degree $2$
Conductor $6026$
Sign $1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.66·3-s + 4-s + 1.81·5-s + 1.66·6-s + 3.25·7-s − 8-s − 0.218·9-s − 1.81·10-s − 4.39·11-s − 1.66·12-s − 0.979·13-s − 3.25·14-s − 3.02·15-s + 16-s + 7.76·17-s + 0.218·18-s + 0.157·19-s + 1.81·20-s − 5.42·21-s + 4.39·22-s − 23-s + 1.66·24-s − 1.70·25-s + 0.979·26-s + 5.36·27-s + 3.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.962·3-s + 0.5·4-s + 0.811·5-s + 0.680·6-s + 1.23·7-s − 0.353·8-s − 0.0729·9-s − 0.573·10-s − 1.32·11-s − 0.481·12-s − 0.271·13-s − 0.869·14-s − 0.781·15-s + 0.250·16-s + 1.88·17-s + 0.0516·18-s + 0.0360·19-s + 0.405·20-s − 1.18·21-s + 0.937·22-s − 0.208·23-s + 0.340·24-s − 0.341·25-s + 0.192·26-s + 1.03·27-s + 0.615·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098410519\)
\(L(\frac12)\) \(\approx\) \(1.098410519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 + T \)
131 \( 1 - T \)
good3 \( 1 + 1.66T + 3T^{2} \)
5 \( 1 - 1.81T + 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 + 4.39T + 11T^{2} \)
13 \( 1 + 0.979T + 13T^{2} \)
17 \( 1 - 7.76T + 17T^{2} \)
19 \( 1 - 0.157T + 19T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 0.703T + 31T^{2} \)
37 \( 1 - 2.06T + 37T^{2} \)
41 \( 1 + 9.75T + 41T^{2} \)
43 \( 1 + 2.90T + 43T^{2} \)
47 \( 1 - 0.804T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 6.46T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 7.40T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.005006528694983462075636365231, −7.57186859872405733408260288257, −6.66700584585494268882633932945, −5.65828382285218087988594226374, −5.45613827043635181004367601133, −4.91195608281835740836652438847, −3.52385395581621238207933707310, −2.44256029623816688506700263028, −1.69556202473483218694640493863, −0.65160046760779705041873330009, 0.65160046760779705041873330009, 1.69556202473483218694640493863, 2.44256029623816688506700263028, 3.52385395581621238207933707310, 4.91195608281835740836652438847, 5.45613827043635181004367601133, 5.65828382285218087988594226374, 6.66700584585494268882633932945, 7.57186859872405733408260288257, 8.005006528694983462075636365231

Graph of the $Z$-function along the critical line